Problem: Simplify and expand the following expression: $ \dfrac{10}{y - 10}+\dfrac{y - 9}{y + 8} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(y - 10)(y + 8)$ Multiply the first term by $\dfrac{y + 8}{y + 8}$ $ \begin{align*} \dfrac{10}{y - 10} \times \dfrac{y + 8}{y + 8} & = \dfrac{(10)(y + 8)}{(y - 10)(y + 8)} \\ & = \dfrac{10y + 80}{(y - 10)(y + 8)}\end{align*} $ Multiply the second term by $\dfrac{y - 10}{y - 10}$ $ \begin{align*} \dfrac{y - 9}{y + 8} \times \dfrac{y - 10}{y - 10} & = \dfrac{(y - 9)(y - 10)}{(y + 8)(y - 10)} \\ & = \dfrac{y^2 - 19y + 90}{(y + 8)(y - 10)}\end{align*} $ Now we have: $ = \dfrac{10y + 80}{(y - 10)(y + 8)} + \dfrac{y^2 - 19y + 90}{(y + 8)(y - 10)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{10y + 80 + y^2 - 19y + 90}{(y - 10)(y + 8)} $ $ = \dfrac{-9y + 170 + y^2}{(y - 10)(y + 8)}$ Expand the denominator: $ = \dfrac{-9y + 170 + y^2}{y^2 - 2y - 80}$